q-Bernstein polynomials associated with q-stirling numbers and Carlitz's q-Bernoulli numbers

Recently, Kim proposed interesting q-extension of Bernstein polynomials and positive linear operators on C[0,1] which are different Phillips' q-Bernstein polynomials. From Kim's q-Bernstein polynomials, we investigate some interesting properties of q-Bernstein polynomials associated with q-stirling numbers and Carlitz's q-Bernoulli numbers


Introduction
Let p be a fixed prime number. Throughout this paper, Z p , Q p , C and C p denote the ring of p-adic integers, the field of p-adic rational numbers, the complex number field and the completion of algebraic closure of Q p , respectively. Let N be the set of natural numbers and Z + = N∪{0}. Let ν p be the normalized exponential valuation of C p with |p| p = p −νp(p) = 1 p . When one talks of q-extension, q is variously considered as an indeterminate, a complex number q ∈ C or p-adic number q ∈ C p . If q ∈ C, one normally assumes |q| < 1, and if q ∈ C p , one normally assumes |1 − q| p < 1.
The q-bosonic natural numbers are defined by [n] q = 1−q n 1−q = 1+q+q 2 +· · ·+q n−1 for n ∈ N, and the q-factorial is defined by [n] q ! = [n] q [n − 1] q · · · [2] q [1] q . For the q-extension of binomial coefficient, we use the following notation in the form of Let C[0, 1] denote the set of continuous functions on [0, 1](⊂ R). Then Bernstein operator for f ∈ C[0, 1] is defined by where n, k ∈ Z + . The polynomials B k,n (x) = n k x k (1 − x) n−k are called Bernstein polynomials of degree n (see [1]). For f ∈ C[0, 1], Kim's q-Bernstein operator of order n for f is defined by 2000 Mathematics Subject Classification : 11B68, 11B73, 41A30. Key words and phrases : q-Bernstein polynomial, Bernoulli numbers and polynomials, p-adic q-integral.
are called the Kim's q-Bernstein polynomials of degree n (see [4]).
We say that f is uniformly differentiable function at a point a ∈ Z p , and write x−y has a limit f ′ (a) as (x, y) → (a, a). For f ∈ U D(Z p ), the p-adic q-integral on Z p is defined by f (x)q x , (see [6]).
The k-th order factorial of the q-number [x] q , which is defined by is called the q-factorial of x of order k (see [6]).
In this paper, we give p-adic q-integral representation for Kim's q-Bernstein polynomials and derive some interesting identities for the Kim's q-Bernstein polynomials associated with q-extension of binomial distribution, q-Stirling numbers and Carlitz's q-Bernoulli numbers.

q-Bernstein polynomials
In this section, we assume that 0 < q < 1.
q | a i ∈ R} be the space of q-polynomials of degree less than or equal to n.
For f ∈ C[0, 1] and n, k ∈ Z + , Kim's q-Bernstein operator of order n for f is defined by Here are the Kim's q-Bernstein polynomials of degree n (see [4]).
Kim's q-Bernstein polynomials of degree n is a basis for the space of q-polynomials of degree less than or equal to n. That is, Kim's q-Bernstein polynomials of degree n is a basis for P q .
We see that Kim's q-Bernstein polynomials of degree n span the space of qpolynomials. That is, any q-polynomials of degree less than or equal to n can be written as a linear combination of the Kim's q-Bernstein polynomials of degree n. For n, k ∈ Z + and x ∈ [0, 1], we have If there exist constants C 0 , C 1 , . . . , C n such that C 0 B 0,n (x, q) + C 1 B 1,n (x, q) + · · · + C n B n,n (x, q) = 0 holds for all x, then we can derive the following equation from (3): Since the power basis is a linearly independent set, it follows that which implies that C 0 = C 1 = · · · = C n = 0 (C 0 is clearly zero, substituting this in the second equation gives C 1 = 0, substituting these two into the third equation gives C 2 = 0, and so on). Let us consider a q-polynomial P q (x) ∈ P q which is written by a linear combination of Kim's q-Bernstein basis functions as follows: It is easy to write (4) as a dot product of two values.
From (5), we can derive the following equation: where the b ij are the coefficients of the power basis that are used to determine the respective Kim's q-Bernstein polynomials. We note that the matrix in this case is lower triangular.
From (2) and (3), we note that In the quadratic case (n = 2), the matrix representation is In the cubic case (n = 3), the matrix representation is In many applications of q-Bernstein polynomials, a matrix formulation for the Kim's q-Bernstein polynomials seems to be useful.

q-Bernstein polynomials, q-Stirling numbers and q-Bernoulli numbers
In this section, we assume that q ∈ C p with |1 − q| p < 1. For f ∈ U D(Z p ), let us consider the p-adic analogue of Kim's q-Bernstein type operator of order n on Z p as follows: Let (Eh)(x) = h(x + 1) be the shift operator. Then the q-difference operator is defined by where (Ih)(x) = h(x). From (6), we derive the following equation: , (see [7]).
The q-Stirling number of the first kind is defined by S 1,q (n, k)z k , (see [5,6]), (8) and the q-Stirling number of the second kind is also defined by S 2,q (n, k)z k , (see [5]).
By (6), (7), (8) and (9), we get for n, k ∈ Z + (see [6]). Let us consider Kim's q-Bernstein polynomials of degree n on Z p as follows: for n, k ∈ Z + and x ∈ Z p . Thus, we easily see that By (1) and (10), we obtain the following proposition.
From the definition of Kim's q-Bernstein polynomial, we note that where i ∈ N. From the definition of q-binomial coefficient, we have By (12), we see that , (see [6,7]).
Theorem 2. For n, k ∈ Z + and i ∈ N, we have It is easy to see that for i ∈ N, By (11) and (14), we easily get , (see [6]).

Thus, we have
Zp By (1) and (15), we obtain the following corollary.
Corollary 3. For n, k ∈ Z + and i ∈ N, we have It is known that and By simple calculation, we have that From (17) and (18), we note that n m = n k=m (q − 1) −m+k n k q S 1,q (k, m), (see [6]).
Thus, we obtain the following proposition.
Proposition 4. For n, k ∈ Z + , we have From the definition of the q-Stirling numbers of the first kind, we get By (11) and (19), we obtain the following theorem.
Theorem 5. For n, k ∈ Z + and i ∈ N, we have By (14) and Theorem 5, we obtain the following corollary.
Corollary 6. For i ∈ Z + , we have The q-Bernoulli polynomials of order k ∈ Z + are defined by Thus, we have [6]).
Theorem 7. For i, n, k ∈ Z + , we have It is easy to show that q ( n 2 ) x n q = 1 [n] q !